27,071 research outputs found
A Subquadratic Bound for Online Bisection
In the online bisection problem one has to maintain a partition of
elements into two clusters of cardinality . During runtime, an online
algorithm is given a sequence of requests, each being a pair of elements: an
inter-cluster request costs one unit while an intra-cluster one is free. The
algorithm may change the partition, paying a unit cost for each element that
changes its cluster.
This natural problem admits a simple deterministic -competitive
algorithm [Avin et al., DISC 2016]. While several significant improvements over
this result have been obtained since the original work, all of them either
limit the generality of the input or assume some form of resource augmentation
(e.g., larger clusters). Moreover, the algorithm of Avin et al. achieves the
best known competitive ratio even if randomization is allowed.
In this paper, we present a first randomized online algorithm that breaks
this natural barrier and achieves a competitive ratio of
without resource augmentation and for an arbitrary sequence of requests
Run Generation Revisited: What Goes Up May or May Not Come Down
In this paper, we revisit the classic problem of run generation. Run
generation is the first phase of external-memory sorting, where the objective
is to scan through the data, reorder elements using a small buffer of size M ,
and output runs (contiguously sorted chunks of elements) that are as long as
possible.
We develop algorithms for minimizing the total number of runs (or
equivalently, maximizing the average run length) when the runs are allowed to
be sorted or reverse sorted. We study the problem in the online setting, both
with and without resource augmentation, and in the offline setting.
(1) We analyze alternating-up-down replacement selection (runs alternate
between sorted and reverse sorted), which was studied by Knuth as far back as
1963. We show that this simple policy is asymptotically optimal. Specifically,
we show that alternating-up-down replacement selection is 2-competitive and no
deterministic online algorithm can perform better.
(2) We give online algorithms having smaller competitive ratios with resource
augmentation. Specifically, we exhibit a deterministic algorithm that, when
given a buffer of size 4M , is able to match or beat any optimal algorithm
having a buffer of size M . Furthermore, we present a randomized online
algorithm which is 7/4-competitive when given a buffer twice that of the
optimal.
(3) We demonstrate that performance can also be improved with a small amount
of foresight. We give an algorithm, which is 3/2-competitive, with
foreknowledge of the next 3M elements of the input stream. For the extreme case
where all future elements are known, we design a PTAS for computing the optimal
strategy a run generation algorithm must follow.
(4) Finally, we present algorithms tailored for nearly sorted inputs which
are guaranteed to have optimal solutions with sufficiently long runs
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